In his monograph Chebyshev and Fourier Spectral Methods, John Boyd claimed that, regarding Fourier spectral methods for solving differential equations, ``[t]he virtues of the Fast Fourier Transform will continue to improve as the relentless march to larger and larger [bandwidths] continues''. This paper attempts to further the virtue of the Fast Fourier Transform (FFT) as not only bandwidth is pushed to its limits, but also the dimension of the problem. Instead of using the traditional FFT however, we make a key substitution: a high-dimensional, sparse Fourier transform (SFT) paired with randomized rank-1 lattice methods. The resulting sparse spectral method rapidly and automatically determines a set of Fourier basis functions whose span is guaranteed to contain an accurate approximation of the solution of a given elliptic PDE. This much smaller, near-optimal Fourier basis is then used to efficiently solve the given PDE in a runtime which only depends on the PDE's data compressibility and ellipticity properties, while breaking the curse of dimensionality and relieving linear dependence on any multiscale structure in the original problem. Theoretical performance of the method is established herein with convergence analysis in the Sobolev norm for a general class of non-constant diffusion equations, as well as pointers to technical extensions of the convergence analysis to more general advection-diffusion-reaction equations. Numerical experiments demonstrate good empirical performance on several multiscale and high-dimensional example problems, further showcasing the promise of the proposed methods in practice.

翻译：在其专著《Chebyshev与Fourier谱方法》中，John Boyd声称，关于求解微分方程的Fourier谱方法，"随着带宽不断向更大范围扩展，快速Fourier变换的优势将持续提升"。本文旨在进一步发扬快速Fourier变换（FFT）的优势，不仅将带宽推向极限，同时处理问题维度的挑战。但我们并非采用传统FFT，而是进行了一项关键替换：将高维稀疏Fourier变换（SFT）与随机秩1格子方法相结合。由此产生的稀疏谱方法能够快速自动确定一组Fourier基函数，其张成空间保证包含给定椭圆型偏微分方程解的精确近似。随后利用这一规模更小、近乎最优的Fourier基高效求解给定偏微分方程，其运行时间仅取决于方程数据的可压缩性与椭圆性，同时打破维数灾难，并消除原始问题中任何多尺度结构的线性依赖性。本文通过Sobolev范数下对一般非恒定扩散方程的收敛性分析，确立了该方法的理论性能，并给出了将收敛分析推广至更一般对流-扩散-反应方程的技术扩展指引。数值实验在多个多尺度与高维示例问题上展现了良好的实证性能，进一步彰显了所提方法在实际应用中的潜力。